Extenics Coordinated Torque Distribution Control for Distributed Drive Electric Vehicles Considering Stability and Energy Efficiency

Abstract

To address the challenges of enhancing driving stability and energy efficiency in distributed-drive electric vehicles, this paper proposes an extenics coordinated torque distribution control method that integrates energy efficiency optimization and vehicle stability. The primary contribution was the development of a vehicle stability assessment method grounded in extenics control theory, which was used to obtain the vehicle’s phase plane and stability region. Subsequently, an objective function with constraints for in-wheel motor torque distribution was formulated, targeting both optimal energy efficiency and maximum tire stability margin. Furthermore, the extension distances from the actual vehicle state to the stability boundaries were computed to determine adaptive weighting coefficients for these dual objectives. Finally, a Matlab/Simulink 2018a and Carsim2019 co-simulation platform was built to implement and test the proposed method. Simulations under the NEDC urban driving cycle and double-lane-change driving conditions were conducted to evaluate the following three distribution strategies: energy-optimal, stability-oriented, and extenics coordinated control. The results demonstrated that, regarding vehicle stability performance, extenics coordinated control showed a slightly inferior performance to the stability-oriented approach but substantially outperformed the energy-optimal strategy. In terms of energy consumption, the energy-optimal strategy achieved the lowest loss and the stability-oriented strategy showed the highest, while the extenics coordinated control presented intermediate results of 5.4 × 10⁹ J and 9.7 × 10⁷ J, respectively, under two driving conditions, representing reductions of 2.17% and 11.2% compared to the stability-oriented method. The proposed torque distribution method establishes an effective balance between energy-optimal and stability-oriented objectives. This method not only ensures satisfactory driving stability, but also reduces energy loss in in-wheel motors.

1. Introduction

With the continuous advancement of vehicle intelligence and by-wire technologies, Distributed-Drive Electric Vehicles (DDEVs) integrate in-wheel motors inside the wheels, featuring independent drive, compact structure, and a high transmission efficiency. These characteristics contribute to enhanced driving stability and energy efficiency. Currently, the most widely adopted torque distribution control strategy for DDEVs is hierarchical control. The upper-level controller calculates generalized forces through direct yaw moment control, while the lower-level controller rationally allocates these generalized forces or torques to individual wheels, thereby ensuring both driving safety and energy efficiency.

Numerous scholars have conducted research on single-objective torque distribution control methods [1,2,3,4]. In the context of torque distribution control aimed at enhancing vehicle stability [5,6,7,8], Aria et al. [9] allocated upper-level control torque to four in-wheel motors based on dynamic tire load transfer and designed a slip-rate-based sliding mode controller to correct the torque. Their simulation results demonstrated a reduction in lateral acceleration, improved tracking of the reference yaw rate, a decrease in the vehicle sideslip angle, and a further enhancement of vehicle handling stability. Ren et al. [10] treated the torque outputs of four in-wheel motors as the control inputs for the MPC system. The objective function of their controller incorporated not only the tracking performance of vehicle states relative to desired references, but also constraints including the maximum torque of in-wheel motors, tire slip rates, torque variation limits, and steering requirements. Their simulation results demonstrated that rational torque allocation enabled satisfactory controller performance, effectively aligning with driver intentions. Shi et al. [11] formulated the torque distribution for individual wheels as a nonlinear constrained optimization problem, with the objective function focused on minimizing longitudinal control error. The solution was obtained using the Sequential Quadratic Programming (SQP) method. Through simulation validation combined with upper-level control, the results demonstrated that the proposed approach enhanced vehicle handling stability during typical lane-change maneuvers and reduced driver workload. Su et al. [12] proposed a torque vectoring control system to enhance the handling stability of distributed-drive electric buses under complex driving conditions. This system adopts a hierarchical control structure, where the upper-level controller consists of a direct yaw moment controller based on feedforward and feedback control and the lower-level controller functions as a torque allocation controller. Comparative real vehicle test results demonstrated that the torque vectoring control system significantly improved the bus’s handling stability. Hu et al. [13] proposed a coordinated torque distribution strategy considering wheel slip to address in-wheel motor torque loss scenarios. By developing an anti-wheel-slip torque constraint mechanism and a distribution strategy accounting for torque loss, coordinated allocation of four-wheel torque was achieved. The effectiveness of the proposed approach was validated through hardware-in-the-loop testing. Huang et al. [14] proposed a hierarchical control system for handling stability in distributed-drive electric vehicles. The upper-level controller employs robust H-infinity control, while the lower-level controller adopts a rule-based torque vectoring allocation strategy. Co-simulation results utilizing Carsim and Matlab demonstrated that the proposed rule-based torque vectoring strategy effectively enhanced both vehicle handling stability and ride comfort.

In terms of achieving optimal energy efficiency as the torque distribution control objective [15,16,17,18], Adeleke et al. [19] proposed an economy-oriented torque distribution strategy. They established a comprehensive model of a distributed-drive electric vehicle and employed the Dynamic Programming (DP) algorithm for torque distribution. Through simulations and hardware-in-the-loop testing, comparisons were made with a fuzzy logic control algorithm. The results demonstrated that the proposed method further reduced vehicle energy consumption and ensured optimal torque distribution. Hu et al. [20] established the maximization of in-wheel motor efficiency as the objective function, with constraints including the maximum torque of the in-wheel motors and the front–rear axle torque distribution ratio. They allocated the target control forces and moments accordingly. This control algorithm achieved a peak motor efficiency of 89.49%, representing a 0.42% improvement over a uniform distribution strategy. Cao et al. [21] investigated dynamic driving conditions of four-wheel-drive electric vehicles and proposed a multi-objective optimal torque distribution strategy that comprehensively considers load transfer between axles and variations in adhesion characteristics. Their simulation results demonstrated that the proposed strategy achieved a significantly better economic performance compared to both the average torque distribution strategy and an optimization strategy based on the sequential quadratic programming algorithm. Sun et al. [22] investigated the cornering scenario of distributed-drive electric vehicles by establishing a seven-degree-of-freedom vehicle dynamics model. They applied Pontryagin’s minimum principle to study torque vectoring allocation during vehicle steering and performed offline optimization of the torque distribution coefficients to reduce energy consumption in cornering conditions. Shangguan et al. [23] proposed a hierarchical control framework for electric buses driven by in-wheel motors. The upper-level controller incorporates a nonlinear observer for estimating longitudinal and lateral velocities, while the lower-level controller employs nonlinear model predictive control. This structure enabled accurate path tracking while minimizing energy consumption of the bus. Tan et al. [24] proposed an energy-saving torque distribution strategy for off-road distributed drive vehicles. By analyzing the characteristics of the motor system, the optimal inter-axle torque distribution scheme was determined to improve energy conversion efficiency. Quadratic programming was employed to optimize the inter-wheel torque allocation. Co-simulation results demonstrated that, compared with conventional methods, the proposed approach not only ensured optimal torque distribution for path tracking, but also significantly enhanced the overall energy efficiency of the distributed-drive vehicle.

A limited number of scholars have investigated torque distribution control methods that simultaneously consider stability and economy. Qiu et al. [25] proposed a fuzzy yaw moment control strategy based on the golden section search algorithm for distributed-drive electric buses. Simulations and hardware-in-the-loop experiments demonstrated that, compared with the four-wheel average distribution strategy, the proposed method improved torque distribution efficiency by approximately 4% under two steering conditions. It also comprehensively enhanced the vehicle’s energy utilization rate while effectively reducing deviations in yaw rate and sideslip angle, thereby significantly improving vehicle stability. Liu et al. [26] proposed a hierarchical torque vectoring control strategy to coordinate maneuver stability and energy consumption in distributed-drive electric vehicles. By integrating road adhesion coefficient estimation, adaptive model predictive control, and stability boundary-based torque distribution, the proposed strategy achieved a 44.88% reduction in the root mean square error of sideslip angle and a 24.45% decrease in motor power consumption during double-lane-change maneuvers compared to conventional torque distribution methods. These results demonstrated a significant improvement in overall vehicle performance. Jia et al. [27] developed a lane-changing control system utilizing pure electric driving range and remaining range, and performed simulations to analyze both the performance of this control model and the economic benefits of the integrated eco-driving strategy.

In summary, existing research predominantly focuses on motor torque distribution control for distributed-drive electric vehicles (DDEVs) with a single objective, either stability or energy efficiency. However, the actual operating environments of these vehicles are complex and variable. Consequently, torque allocation must not only ensure driving stability, but also address energy dissipation and driving range requirements. It is, therefore, imperative to develop integrated torque distribution control methodologies that simultaneously optimize both vehicle stability and energy efficiency.

In response to the above issues, this study proposes a torque distribution methodology for distributed-drive electric vehicles based on extenics coordinated control, with consideration given to both vehicle stability and energy efficiency. This study innovatively constructs a torque distribution objective function aimed at achieving optimal energy efficiency and maximum tire stability margin. By utilizing extenics control theory to adaptively coordinate the weight coefficients, a torque distribution control method based on extenics coordinated control theory is ultimately realized. Its effectiveness is validated through a Carsim/Simulink co-simulation platform.

2. Materials and Methods

2.1. Two-Degree-of-Freedom Mathematical Model of Vehicle Dynamics

2.1.1. Nonlinear Two-Degree-of-Freedom Vehicle Dynamics Model

The vehicle stability investigated in this study primarily concerns the planar motion parallel to the ground. Therefore, only the following two degrees of freedom are considered: the yaw motion and the lateral motion of the vehicle. The two-degree-of-freedom vehicle dynamics model (bicycle model) is illustrated in Figure 1. In the figure, oxy represents the vehicle coordinate system; F_xf and F_xr denote the total longitudinal forces of the front and rear wheels, respectively; F_yf and F_yr denote the total lateral forces of the front and rear wheels, respectively; α_f and α_r represent the tire sideslip angles of the front and rear wheels, respectively; v is the vehicle travel speed; v_x is the vehicle longitudinal speed; v_y is the vehicle lateral speed; β is the vehicle sideslip angle at the center of mass; γ is the vehicle yaw rate; δ is the front-wheel steering angle; and L_f and L_r are the distances from the front and rear axles to the vehicle’s center of mass, O, respectively.

The differential equations of the two-degree-of-freedom vehicle dynamics model are as follows:

$$\left\{\begin{array}{l}m({\dot{v}}_y+v_x\gamma )=F_{xf}\sin \delta +F_{yf}\cos \delta +F_{yr}\\ I_z\dot{\gamma }=L_f\left(F_{xf}\sin \delta +F_{yf}\cos \delta \right)-L_r{F}_{yr}\end{array}\right.$$

(1)

During cornering maneuvers, the front-wheel steering angle is assumed to be a small angle, namely $\cos \delta \approx 1$, $\sin \delta \approx \delta$, so the following result is established:

$$\left\{\begin{array}{l}m({\dot{v}}_y+v_x\gamma )=F_{xf}\delta +F_{yf}+F_{yr}\\ I_z\dot{\gamma }=L_f\left(F_{xf}\delta +F_{yf}\right)-L_r{F}_{yr}\end{array}\right.$$

(2)

Given that the vehicle sideslip angle β is assumed to be small, it follows that $v_y=v\sin \beta =v\beta$, $v_x=v\cos \beta =v$. Based on the kinematic geometry of the vehicle and the definition of the tire slip angle, the slip angles of the front and rear tires are derived as follows:

$$\left\{\begin{array}{l}{\alpha }_f={\displaystyle \frac{v\beta +L_f\gamma }{v}}-\delta \\ {\alpha }_r={\displaystyle \frac{v\beta -L_r\gamma }{v}}\end{array}\right.$$

(3)

Based on tire cornering characteristics, when the tire slip angle becomes excessively large, the tire operates in the nonlinear region, causing the vehicle to exhibit nonlinear behavior during actual driving. Therefore, when analyzing vehicle handling and stability, selecting the magic formula, which effectively captures the dynamic characteristics of tires, is sufficient to meet the requirements.

For a given initial vertical load and camber angle, the general expression of the magic formula is as follows [28]:

$$\left\{\begin{array}{l}y=D\sin \left\{C\arctan \left[Bx-E\left(Bx-\arctan (Bx)\right)\right]\right\}\\ Y(X)=y(x)+S_V\\ x=X+S_H\end{array}\right.$$

(4)

The physical quantities represented by each symbol in Equation (4) are explained in

Table 1. Physical quantities in the magic formula tire model.

Notation	Physical Quantity
Y	Output variable, representing the tire longitudinal force Fx, lateral force Fy, or aligning moment Mz
X	Input variable, representing the wheel slip angle or longitudinal slip ratio
B	Stiffness factor, determining the slope at the origin
C	Shape factor, influencing the form of the curve and defining the effective range of the sine function
D	Peak factor, controlling the maximum value of the curve
E	Curvature factor, adjusting the curvature near the peak and influencing the horizontal position of the maximum value
SV	Vertical shift in the curve relative to the origin
SH	Horizontal shift in the curve relative to the origin

. The tire model parameters can be obtained through fitting.

Under combined slip conditions, the tire longitudinal force, lateral force, and aligning moment are expressed as follows:

$$\left\{\begin{array}{l}{F}_x=F_{x0}{G}_{x\alpha }(\alpha ,\lambda ,\gamma ,F_z)\\ F_y=F_{y0}{G}_{y\lambda }(\alpha ,\lambda ,\gamma ,F_z)+S_{Vy\lambda }\\ M_z=-t{F^{\prime }}_y+M_{zr}\end{array}\right.$$

(5)

where F_x0 and F_y0 represent the tire longitudinal force and lateral force under pure longitudinal slip and pure side slip conditions, respectively; S_vyλ is the drift factor present when fitting the tire lateral force; M_zr is the residual aligning moment of the tire; ${F^{\prime }}_y=F_y-F_{Vy\lambda }$; and $G_{x\alpha }(\alpha ,\lambda ,\gamma ,F_z)$ and $G_{y\lambda }(\alpha ,\lambda ,\gamma ,F_z)$ are weighting coefficients for the longitudinal force and lateral force under combined conditions, which are functions of vertical load, slip angle, slip ratio, and camber angle.

2.1.2. Two-Degree-of-Freedom Linear Vehicle Dynamics Model

In the analysis of vehicle handling and stability, under the assumptions that the vehicle operates at a constant speed and the tire cornering characteristics remain within their linear region, the lateral forces generated by the front and rear tires are given by the following:

$$\left\{\begin{array}{l}{F}_{yf}=k_f{\alpha }_f\\ F_{yr}=k_r{\alpha }_r\end{array}\right.$$

(6)

where k_f and k_r are the cornering stiffnesses of the front and rear tires, respectively.

This paper neglects longitudinal tire forces by setting F_xf = 0. By combining Equations (2), (3), and (6), we obtain the following:

$$\left\{\begin{array}{l}\dot{\beta }={\displaystyle \frac{k_f+k_r}{mv}}\beta +\left({\displaystyle \frac{L_f{k}_f-L_r{k}_r}{m{v}^2}}-1\right)\gamma -{\displaystyle \frac{k_f}{mv}}\delta \\ \dot{\gamma }={\displaystyle \frac{L_f{k}_f-L_r{k}_r}{I_z}}\beta +{\displaystyle \frac{L_f^2{k}_f+L_r^2{k}_r}{I_{z}v}}\gamma -{\displaystyle \frac{L_f{k}_f}{I_z}}\delta \end{array}\right.$$

(7)

2.1.3. Reference Model

Setting $\dot{\beta }=0$ and $\dot{\gamma }=0$, the vehicle yaw rate under steady-state conditions is given by the following:

$$\gamma ={\displaystyle \frac{v/L}{K^{\prime }{v}^2+1}}\delta$$

(8)

where $K^{\prime }=m\left(L_f/k_r-L_r/k_f\right)/L^2$ is the stability factor, a key parameter for characterizing the vehicle steady-state response.

In the study of vehicle handling and stability, the reference model for the control strategy is defined as follows. To ensure path-tracking capability, the ideal sideslip angle at the vehicle’s center of gravity is set to 0 rad, reflecting the objective of tracking a desired steady-state value. Meanwhile, the yaw rate under steady-state conditions is designated as the target to preserve handling stability. Furthermore, the road adhesion coefficient must also be incorporated. The reference model for the vehicle handling stability control strategy is, thus, formulated as follows:

$$\left\{\begin{array}{l}{\beta }_d=0\\ {\gamma }_d=\min \left(\left|{\displaystyle \frac{\delta v/L}{K^{\prime }{v}^2+1}}\right|,{\displaystyle \frac{0.85\mu g}{v}}\right)\mathrm{sgn}\left(\delta \right)\end{array}\right.$$

(9)

where μ is the road adhesion coefficient.

2.2. Research on Vehicle Motion Control Methods

2.2.1. Longitudinal Speed Tracking Control

To ensure the accuracy of vehicle speed tracking, this paper introduces fractional-order operators ${}_{t_0}D_{\alpha }^t$ and designs a fractional-order PID controller based on fractional-order calculus theory [29,30,31]. The desired vehicle speed v_obj is set as the control target for the longitudinal vehicle speed. A fractional-order PID (FOPID) controller is devised based on the error between the actual vehicle speed v and v_obj. This controller yields the required total longitudinal drive torque, thereby enabling the vehicle to track the desired speed. The flowchart of this longitudinal speed control process is illustrated in Figure 2.

2.2.2. Vehicle Lateral Stability Control

To fulfill the requirement for stable vehicle handling performance, the linear two-degree-of-freedom vehicle dynamics model in this paper is augmented with the target yaw moment, ΔM, and the lateral force, F_dy. This augmented model can be transformed from Equation (7) into Equation (10), as follows:

$$\left\{\begin{array}{l}\dot{\beta }={\displaystyle \frac{k_f+k_r}{mv}}\beta +{\displaystyle \frac{L_f{k}_f-L_r{k}_r-m{v}^2}{m{v}^2}}\gamma -{\displaystyle \frac{k_f}{mv}}\delta +{\displaystyle \frac{F_{dy}}{mv}}\\ \dot{\gamma }={\displaystyle \frac{L_f{k}_f-L_r{k}_r}{I_z}}\beta +{\displaystyle \frac{L_f^2{k}_f+L_r^2{k}_r}{I_{z}v}}\gamma -{\displaystyle \frac{L_f{k}_f}{I_z}}\delta +{\displaystyle \frac{△M}{I_z}}\end{array}\right.$$

(10)

By setting $e={\left[e_{\gamma }\hspace{1em}{e}_{\beta }\right]}^T$, the integral sliding mode surface is defined as follows:

$$s_I={\lambda }_I{\displaystyle {\int }_0^{t}e}dt+e$$

(11)

where $s_I={\left[s_{\gamma }\hspace{1em}{s}_{\beta }\right]}^T$; $e_{\gamma }={\gamma }_d-\gamma$; $e_{\beta }={\beta }_d-\beta$; ${\lambda }_I=diag({\lambda }_{\gamma },{\lambda }_{\beta })$, λ_γ and λ_β are positive real numbers.

The exponential reaching law is selected as follows:

$${\dot{s}}_I=-k_{I}s-{\epsilon }_I\mathrm{sgn}(s_I)$$

(12)

where both $k_I=diag\left(k_{\gamma },k_{\beta }\right)$ and ${\epsilon }_I=diag\left({\epsilon }_{\gamma },{\epsilon }_{\beta }\right){\displaystyle \frac{1}{2}}$ are positive real matrices.

Differentiating Equation (11) yields the following:

$${\dot{s}}_I={\lambda }_{I}e+\dot{e}$$

(13)

By combining Equations (10), (12), and (13), the control laws for the target yaw moment, ΔM, and the lateral force, F_dy, are derived as follows:

$$\left\{\begin{array}{l}\mathrm{\Delta }M=I_z\left[{\dot{\gamma }}_d+{\lambda }_{\gamma }{e}_{\gamma }-{\displaystyle \frac{L_f{k}_f-L_r{k}_r}{I_z}}\beta -{\displaystyle \frac{L_f^2{k}_f+L_r^2{k}_r}{I_{z}v}}\gamma +{\displaystyle \frac{L_f{k}_f}{I_z}}\delta +k_{\gamma }{s}_{\gamma }+{\epsilon }_{\gamma }\mathrm{sgn}(s_{\gamma })\right]\\ F_{dy}=mv\left[{\dot{\beta }}_d+{\lambda }_{\beta }{e}_{\beta }-{\displaystyle \frac{k_f+k_r}{mv}}\beta -{\displaystyle \frac{L_f{k}_f-L_r{k}_r-m{v}^2}{m{v}^2}}\gamma +{\displaystyle \frac{k_f}{mv}}\delta +k_{\beta }{s}_{\beta }+{\epsilon }_{\beta }\mathrm{sgn}(s_{\beta })\right]\end{array}\right.$$

(14)

Construct a Lyapunov function

$$V_I={\displaystyle \frac{1}{2}}{s}_{\gamma }^2+{\displaystyle \frac{1}{2}}{s}_{\beta }^2$$

(15)

Differentiating Equation (15) yields the following:

$${\dot{V}}_I={\dot{s}}_{\gamma }{s}_{\gamma }+{\dot{s}}_{\beta }{s}_{\beta }=({\lambda }_{\gamma }{e}_{\gamma }+{\dot{\gamma }}_d-\dot{\gamma })s_{\gamma }+({\lambda }_{\beta }{e}_{\beta }+{\dot{\beta }}_d-\dot{\beta })s_{\beta }$$

(16)

Substituting Equations (10) and (14) into Equation (16) and simplifying yields the following:

$$\begin{array}{l}{\dot{V}}_I=-\left[k_{\gamma }{s}_{\gamma }+{\epsilon }_{\gamma }\mathrm{sgn}(s_{\gamma })\right]s_{\gamma }-\left[k_{\beta }{s}_{\beta }+{\epsilon }_{\beta }\mathrm{sgn}(s_{\beta })\right]s_{\beta }\\ \hspace{1em}\le -k_{\gamma }{s}_{\gamma }^2-{\epsilon }_{\gamma }\left|s_{\gamma }\right|-k_{\beta }{s}_{\beta }^2-{\epsilon }_{\beta }\left|s_{\beta }\right|\end{array}$$

(17)

Since k_γ, k_β, ε_γ, and ε_β are all positive real numbers, it follows that ${\dot{V}}_I\le 0$. According to the Lyapunov stability theorem, the control laws for the target yaw moment and lateral force ensure the asymptotic stability of the control system.

2.3. Vehicle Stability Judgment Based on Extenics Control Theory

Based on a 2-DOF nonlinear dynamic vehicle model, the characteristic parameters of vehicle handling stability are analyzed to determine the influence of factors such as vehicle speed, road adhesion coefficient, and steering wheel angle on vehicle stability.

In vehicle stability control research, it is common practice to assume a constant vehicle speed as a prerequisite for stability analysis. Neglecting the effects of longitudinal tire forces, Equation (2) can be rewritten as follows:

$$\left\{\begin{array}{l}\dot{\beta }={\displaystyle \frac{F_{yf}+F_{yr}}{mv}}-\gamma \\ \dot{\gamma }={\displaystyle \frac{L_f{F}_{yf}-L_r{F}_{yr}}{I_z}}\end{array}\right.$$

(18)

Without consideration of vehicle dynamic load transfer, the vertical loads on the front and rear axles are as follows:

$$\left\{\begin{array}{l}{F}_{zf}={\displaystyle \frac{mg{L}_r}{L}}\\ F_{zr}={\displaystyle \frac{mg{L}_f}{L}}\end{array}\right.$$

(19)

Based on Equations (3), (18), (19), and the magic formula tire model, a 2-DOF nonlinear dynamic vehicle model is established.

By employing the phase plane method, the phase portraits of sideslip angle and its rate are obtained. The phase trajectories of the vehicle’s state points are analyzed, thereby determining the stable and unstable regions of vehicle motion.

Under various conditions of vehicle speed, road adhesion coefficient, and steering wheel angle, the vehicle’s driving state is assessed using the phase plane method. Research on vehicle control is conducted based on the $\beta -\dot{\beta }$ phase plane. Depending on the vehicle state, the control domain is partitioned into the classical domain, extensible domain, and non-domain using the limit cycle method [32,33].

2.3.1. Determination of the Linear Boundary Parameters for the $\beta -\dot{\beta }$ Phase Plane Stability Region

Assuming good road surface adhesion conditions and a constant vehicle speed, with a steering wheel angle of 0 rad, the phase trajectories corresponding to different initial values γ₀ β₀ and $\beta -\dot{\beta }$ were obtained based on the 2-DOF nonlinear vehicle dynamics model, as shown in Figure 3. The diamond markers in the figure represent the stability region determined by the five-parameter diamond method, with the corresponding parameters provided in

Table 2. Parameters of the five-parameter diamond method.

Parameter	Abscissa $\mathit{\beta }$/(Rad)	Ordinate $\dot{\mathit{\beta }}$/(Rad/s)
Left endpoint	${\beta }_{\lim }^{-}$	0
Right endpoint	${\beta }_{\lim }^{+}$	0
Top vertex	0	${\dot{\beta }}_{\lim }^{+}$
Bottom vertex	0	${\dot{\beta }}_{\lim }^{-}$
Stable point	0	0

.

The approximate expressions for the vehicle sideslip angle and its rate, which satisfy the vehicle stability condition, are given by the following [34]:

$$\left|c_1\beta +c_2\dot{\beta }\right|<1$$

(20)

where c₁ and c₂ are constants that depend on the vehicle state parameters.

It follows that the boundary lines of the diamond-shaped stability region satisfy $\left|c_{1i}{\beta }_i+c_{2i}{\dot{\beta }}_i\right|=1$ for i = 1~4. Substituting the endpoint and vertex parameters from Table 2 into Equation (20) yields the parameters c_1i and c_2i for these four lines.

2.3.2. Acquisition of Stability Region Boundary Based on Extenics Control Theory

(1)

Determination of the classical domain boundary

The ellipse obtained by the limit cycle method is defined as the stability boundary for vehicle motion, where the interior of the ellipse represents the vehicle stability region, i.e., the classical domain of the extension set. By default, the four lines of the diamond boundary are tangent to the ellipse, leading to the following system of equations:

$$\left\{\begin{array}{l}{\left({\displaystyle \frac{O_{a\_in}-\beta }{R_{a\_in}}}\right)}^2+{\left({\displaystyle \frac{O_{b\_in}-\dot{\beta }}{R_{b\_in}}}\right)}^2=1\\ \left|c_1\beta +c_2\dot{\beta }\right|=1\end{array}\right.$$

(21)

where O_{a_in} and O_{b_in} are the coordinates of the center of the inscribed ellipse, corresponding to the horizontal and vertical directions, and R_{a_in}, R_{b_in} define its semi-minor and semi-major axes, respectively.

It can be obtained from Equation (21) that the relationship between the diamond boundary and the ellipse satisfies the following:

$$c_{1i}^2{R}_{a\_in}^2+c_{2i}^2{R}_{b\_in}^2={\left(1-O_{a\_in}{c}_{1i}-O_{b\_in}{c}_{2i}\right)}^2$$

(22)

The inscribed ellipse of the diamond, as shown in Figure 4, is obtained by solving for the semi-major axis, semi-minor axis, and center coordinates using the fmincon function in Matlab. The objective function is the area of the inscribed ellipse, subject to the following equation:

$$J_{interior}=\min (-\pi R_1{R}_2)$$

(23)

Subject to the following:

$$\left\{\begin{array}{l}{c}_{1i}^2{R}_1^2+c_{2i}^2{R}_2^2={\left(1-a_1{c}_{1i}-b_1{c}_{2i}\right)}^2\\ {\beta }_{\lim }^{-}<a_1<{\beta }_{\lim }^{+}\\ {\dot{\beta }}_{\lim }^{-}<b_1<{\dot{\beta }}_{\lim }^{+}\\ 0<R_1<\left({\beta }_{\lim }^{+}-{\beta }_{\lim }^{-}\right)/2\\ 0<R_2<\left({\dot{\beta }}_{\lim }^{+}-{\dot{\beta }}_{\lim }^{-}\right)/2\end{array}\right.$$

(24)

where I = 1~4.

(2)

Acquisition of extension domain boundaries

The circumscribed circle of the rhombus boundary is defined as the extension domain of the extension set. The equation of the circumscribed ellipse is as follows:

$${\left({\displaystyle \frac{O_{a\_ex}-\beta }{R_{a\_ex}}}\right)}^2+{\left({\displaystyle \frac{O_{b\_ex}-\dot{\beta }}{R_{b\_ex}}}\right)}^2=1$$

(25)

where O_{a_ex} and O_{b_ex} represent the x-coordinate and y-coordinate of the circumscribed ellipse’s center, and R_{a_ex} and R_{b_ex} represent its semi-minor axis and semi-major axis, respectively.

The four vertices of the rhombus, (${\beta }_{\lim }^{-}$, 0), (0, ${\dot{\beta }}_{\lim }^{+}$), (${\beta }_{\lim }^{+}$, 0), and (0, ${\dot{\beta }}_{\lim }^{-}$), lie on the circumscribed ellipse. Substituting these coordinates into Equation (25) yields a system of four equations. Solving this system gives the semi-major axis, semi-minor axis, and center of the ellipse, as illustrated in Figure 5.

(3)

Acquisition of extension sets

The circumscribed ellipse and inscribed ellipse of the rhombus are combined to obtain the extension set of the control domain, as shown in Figure 6. The area inside the black dashed line represents the classical domain, where the vehicle remains stable and adopts a torque distribution strategy for the in-wheel motors based on energy efficiency optimization. The region between the black dashed line and the red solid line constitutes the extension domain, where the vehicle operates between the stable and unstable zones. In this region, an extensible combined torque distribution strategy for the in-wheel motors is applied, considering both energy efficiency optimization and stability. The area outside the red solid line is the non-domain, indicating that the vehicle is in an unstable state. Here, the torque distribution strategy for the in-wheel motors is solely based on vehicle stability.

2.4. Extenics Coordinated Torque Distribution Control Method Based on Energy Efficiency Optimization and Stability

2.4.1. Objective Function and Constraints Based on Energy-Optimal Distribution

The efficiency of an individual in-wheel motor was experimentally determined by Elaphe through measurements under various torque and speed conditions, covering both traction and regeneration scenarios. Power losses in the in-wheel motor include electromagnetic and windage losses, losses in the inverter, and mechanical losses within the hub assembly.

Chatzikomis et al. established the following expression for the power loss of a single hub motor [35]:

$$P_{loss}(T_q,n)=\left\{\begin{array}{l}{T}_{q}n\left[1/\eta (T_q,n)-1\right]/9.55\hspace{1em}{T}_q>0\\ T_{q}n\left[\eta (T_q,n)-1\right]/9.55\hspace{1em}{T}_q<0\\ P_{em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_q=0\end{array}\right.$$

(26)

where T_q is the torque of a single in-wheel motor; n is the rotational speed; η(T_q,n) is the efficiency of the in-wheel motor; and P_em is the power loss of the motor at 0 N·m.

Based on the power loss of a single in-wheel motor, the power loss curves under different torque and speed conditions are obtained, as shown in Figure 7. At a constant speed, the power loss increases monotonically with the increase in torque. Under the same torque condition, the power loss increases with a higher speed, particularly at low torque levels where the power loss exhibits a more significant rise as speed increases.

Based on the power loss of a single in-wheel motor, the objective function for torque distribution control of in-wheel motors aimed at optimal energy efficiency is as follows:

$$J_e=\min P_{loss}=\min \left({\displaystyle \sum _{i=f}^r{\displaystyle \sum _{j=l}^r{P}_{loss,ij}}}\right)$$

(27)

where P_loss,ij is the power loss of each in-wheel motor.

Based on the relationship between the target control torque/force and the in-wheel motor torques, the following is derived:

$$F_v=B_v{u}_v$$

(28)

where F_v is the target control torque and force, respectively, $F_v={[\begin{array}{ccc}{T}_x& F_{dy}& \mathrm{\Delta }M\end{array}]}^T$; u_v is the in-wheel motor torque, $u_v={\left[\begin{array}{cccc}{T}_{xfl}& T_{xfr}& T_{xrl}& T_{xrr}\end{array}\right]}^T$; and B_v is the coefficient matrix.

$$B_v=\left[\begin{array}{cccc}\cos \delta & \cos \delta & 1& 1\\ \sin \delta /R_{fl}& \sin \delta /R_{fr}& 0& 0\\ \left(\begin{array}{l}-B_f\cos \delta \\ +2L_f\sin \delta \end{array}\right)/2R_{fl}& \left(\begin{array}{l}{B}_f\cos \delta \\ +2L_f\sin \delta \end{array}\right)/2R_{fr}& {\displaystyle \frac{-B_r}{2R_{rl}}}& {\displaystyle \frac{B_r}{2R_{rr}}}\end{array}\right]$$

, where R_fl and R_fr are the tire radii of the front-left and front-right wheels, respectively; B_f and B_r are the front and rear wheel tracks, respectively.

Due to the torque output limits of the in-wheel motors and the road adhesion coefficient limit, the constraints are formulated as follows:

$$\left|T_{xij}\right|<\min \left({\mu }_{ij}{R}_{ij}{F}_{zij},T_{\max }\right)$$

(29)

where T_xij is the torque of each in-wheel motor.

The quadratic programming-based optimal control allocation algorithm is selected to solve for the in-wheel motor torques. This method distributes the target control moments and forces to the four in-wheel motors, meeting the torque distribution requirements while ensuring optimal energy efficiency of the motors.

Based on the objective function and constraints for energy-efficient torque distribution control of in-wheel motors, a standard quadratic programming problem can be formulated as follows:

$$\left\{\begin{array}{l}{J}_E={\displaystyle \frac{1}{2}}{\left(B_v{u}_v-F_v\right)}^T{W}_v\left(B_v{u}_v-F_v\right)+{\displaystyle \frac{1}{2}}{u}_v^T{W}_{uc}{u}_v\\ st.\hspace{1em}{A}_{eq}{u}_v=b_{eq}\\ \hspace{1em}\hspace{1em}{u}_{v\min }\le u_v\le u_{v\max }\end{array}\right.$$

(30)

where $W_{vc}=diag(1,\hspace{0.33em}1,\hspace{0.33em}1)$; $A_{eq}=B_v$; $b_{eq}=F_v$; $u_{v\max }=-u_{v\min }=\min \left({\mu }_{ij}{R}_{ij}{F}_{zij},T_{\max }\right)$; $W_{uc}=\left[\begin{array}{cccc}{W}_{ucfl}^2& 0& 0& 0\\ 0& W_{ucfr}^2& 0& 0\\ 0& 0& W_{ucrl}^2& 0\\ 0& 0& 0& W_{ucrr}^2\end{array}\right]$, $W_{ucij}=\left\{\begin{array}{l}{n}_{ij}\left[1/\eta (T_{xij},n_{ij})-1\right]/9.55\hspace{1em}\hspace{0.33em}{T}_{xij}>0\\ n_{ij}\left[\eta (T_{x_{ij}},n_{ij})-1\right]/9.55\hspace{1em}\hspace{1em}{T}_{xij}<0\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}{T}_{xij}=0\end{array}\right.$.

2.4.2. Objective Function and Constraints Based on Stability-Oriented Distribution

For the torque distribution control strategy of in-wheel motors on complex road surfaces, the objective function and constraints are formulated based on maximizing the tire stability margin, road adhesion conditions, and motor operational characteristics. Furthermore, the vehicle roll angle is incorporated as an additional constraint to enhance safety and stability under extreme driving conditions.

Vehicle body roll induces a shift in the center of mass, which reduces the vehicle’s inherent ability to resist rollover through its own weight and decreases the rollover threshold. Once the roll angle exceeds the critical threshold, vehicle rollover occurs, leading to severe personal injury and property loss. The Lateral-Load Transfer Rate (LTR), a widely adopted rollover indicator, can effectively identify the transient rollover state of a vehicle. Its expression is given as follows:

$$LTR={\displaystyle \frac{\left|F_{zfl}+F_{zrl}-F_{zfr}-F_{zrr}\right|}{F_{zfl}+F_{zrl}+F_{zfr}+F_{zrr}}}$$

(31)

where the LTR value lies within the range of [0, 1].

Neglecting road gradient and considering vehicle load transfer, the dynamic vertical load exerted by the ground on the tires is expressed as follows:

$$\left\{\begin{array}{l}{F}_{zfl}={\displaystyle \frac{mg{L}_r}{2L}}-{\displaystyle \frac{m{a}_x{h}_g}{2L}}-{\displaystyle \frac{m{a}_y{h}_g{L}_r}{B_{f}L}}-{\displaystyle \frac{mg{h}_g{L}_r\sin \phi }{B_{f}L}}\\ F_{zfr}={\displaystyle \frac{mg{L}_r}{2L}}-{\displaystyle \frac{m{a}_x{h}_g}{2L}}+{\displaystyle \frac{m{a}_y{h}_g{L}_r}{B_{f}L}}+{\displaystyle \frac{mg{h}_g{L}_r\sin \phi }{B_{f}L}}\\ F_{zrl}={\displaystyle \frac{mg{L}_f}{2L}}+{\displaystyle \frac{m{a}_x{h}_g}{2L}}-{\displaystyle \frac{m{a}_y{h}_g{L}_f}{B_{r}L}}-{\displaystyle \frac{mg{h}_g{L}_f\sin \phi }{B_{r}L}}\\ F_{zrr}={\displaystyle \frac{mg{L}_f}{2L}}+{\displaystyle \frac{m{a}_y{h}_g{L}_f}{B_{r}L}}+{\displaystyle \frac{m{a}_x{h}_g}{2L}}+{\displaystyle \frac{mg{h}_g{L}_f\sin \phi }{B_{r}L}}\end{array}\right.$$

(32)

where a_x represents the vehicle longitudinal acceleration, a_y represents the vehicle lateral acceleration, h_g represents the height of the vehicle’s center of mass, and φ represents the vehicle roll angle.

Substituting Equation (32) into Equation (31) yields the following:

$$LTR={\displaystyle \frac{\frac{2m{h}_g}{L}\left(L_r/B_f+L_f/B_r\right)\left|a_y+g\sin \phi \right|}{mg}}$$

(33)

As indicated by Equation (33), the LTR is related to both lateral acceleration and vehicle roll angle. When LTR = 0, it indicates no vehicle roll has occurred. When LTR = 1, it signifies that the tires on one side of the vehicle have lifted off the road surface, and rollover is imminent. For LTR values within the interval (0, 1), the metric characterizes the extent of load transfer—the closer the value is to 1, the higher the propensity for vehicle rollover.

Substituting $a_y={\displaystyle \frac{1}{m}}\left({\displaystyle \frac{T_{xfl}}{R_{fl}}}+{\displaystyle \frac{T_{xfr}}{R_{fr}}}\right)\sin \delta +{\displaystyle \frac{F_{yfl}+F_{yfr}}{m}}\cos \delta +{\displaystyle \frac{F_{yrl}+F_{yrr}}{m}}$ into Equation (33) yields the following:

$$LTR={\displaystyle \frac{\left|\frac{2h_g}{L}\left(\frac{L_r}{B_f}+\frac{L_f}{B_r}\right)\left[\begin{array}{l}\left(\frac{T_{xfl}}{R_{fl}}+\frac{T_{xfr}}{R_{fr}}\right)\sin \delta +\left(F_{yfl}+F_{yfr}\right)\cos \delta \\ +F_{yrl}+F_{yrr}+mg\sin \phi \end{array}\right]\right|}{mg}}$$

(34)

The objective function for the stability-oriented torque distribution control of in-wheel motors is derived as follows:

$$\left\{\begin{array}{l}\min f_s={\displaystyle \frac{1}{2}}{\left(B_v{u}_v-F_v\right)}^T{W}_{sv}\left(B_v{u}_v-F_v\right)+{\displaystyle \frac{1}{2}}{u}_v^T{W}_{sc}{u}_v\\ st.\hspace{1em}\hspace{1em}A{u}_v<b\\ \hspace{1em}\hspace{1em} \hspace{1em}{A}_{eq}{u}_v=b_{eq}\\ \hspace{1em}\hspace{1em}{u}_{v\min }\le u_v\le u_{v\max }\end{array}\right.$$

(35)

where $W_{sv}=diag(1,\hspace{0.33em}1,\hspace{0.33em}1)$; $W_{sc}=diag\left(\begin{array}{cccc}{\displaystyle \frac{1}{{\left({\mu }_{fl}{R}_{fl}{F}_{zfl}\right)}^2}},& {\displaystyle \frac{1}{{\left({\mu }_{fr}{R}_{fr}{F}_{zfr}\right)}^2}},& {\displaystyle \frac{1}{{\left({\mu }_{rl}{R}_{rl}{F}_{zrl}\right)}^2}},& {\displaystyle \frac{1}{{\left({\mu }_{rr}{R}_{rr}{F}_{zrr}\right)}^2}}\end{array}\right)$; $A=\left[\begin{array}{cccc}\begin{array}{l}\sin \delta /R_{fl}\\ -\sin \delta /R_{fl}\end{array}& \begin{array}{l}\sin \delta /R_{fr}\\ -\sin \delta /R_{fr}\end{array}& \begin{array}{l}0\\ 0\end{array}& \begin{array}{l}0\\ 0\end{array}\end{array}\right]$; $b=\left[\begin{array}{l}{\displaystyle \frac{mgL}{2h_g\left(L_r/B_f+L_f/B_r\right)}}-\left(F_{yfl}+F_{yfr}\right)\cos \delta -F_{yrl}-F_{yrr}-mg\sin \phi \\ {\displaystyle \frac{mgL}{2h_g\left(L_r/B_f+L_f/B_r\right)}}+\left(F_{yfl}+F_{yfr}\right)\cos \delta +F_{yrl}+F_{yrr}+mg\sin \phi \end{array}\right]$.

2.4.3. Determination of Weighting Coefficients

The two characteristic quantities, sideslip angle and its angular velocity, are selected. The calculation methods for the extension distance and its correlation function are then determined based on the extenics theory approach.

(1)

Computation of the correlation function

In a two-dimensional extension set, the origin O (0, 0) is the optimal point, representing the stable state point of a vehicle. Assuming the vehicle’s arbitrary driving state is denoted as point P, the straight line OP is drawn and extended. This line intersects the boundaries of the classical domain and the extension domain at points labeled sequentially from left to right as P₁, P₂, P₃, and P₄, as illustrated in Figure 8. The segment OP represents the shortest distance for point P to approach the origin O (0, 0). In extension set theory, the extension distance, which is defined within a one-dimensional coordinate system, refers to the distance from a point to a set. Consequently, it is necessary to transform the two-dimensional extension set into a one-dimensional form.

As shown in Figure 8, the extension distance from point P to the classical domain $X_1= <P_2,P_3>$ is defined as $\rho (P,X_1)$, as follows:

$$\rho \left(P,X_1\right)=\left\{\begin{array}{l}\left|P{P}_2\right|\hspace{1em}\hspace{1em}P\in <-\infty ,P_2>\\ -\left|P{P}_2\right|\hspace{1em}P\in <P_2,0>\\ -\left|P{P}_3\right|\hspace{1em}P\in <0,P_3>\\ \left|P{P}_3\right|\hspace{1em}\hspace{1em}P\in <P_3,+\infty >\end{array}\right.\hspace{1em}$$

(36)

Similarly, the extension distance from point P to the extension domain $X_2= <P_1,P_2>\cup <P_3,P_4>$ is $\rho (P,X_2)$; thus, the following correlation function is obtained:

$$K(s)={\displaystyle \frac{\rho (P,<P_1,P_4>)}{D(P,<P_1,P_4>,<P_2,P_3>)}}$$

(37)

where $D(P,<P_1,P_4>,<P_2,P_3>)=\rho (P,<P_1,P_4>)-\rho (P,<P_2,P_3>)$.

(2)

Measurement mode partition and weighting coefficients

Based on the results of the correlation function in Equation (37), the corresponding torque distribution strategy is applied, as follows:

When both β and $\dot{\beta }$ reside within the classical domain, the measurement mode is set to $M_1=\left\{s\right|K(s)\ge 1\}$. Consequently, the torque distribution weight for energy efficiency optimization is assigned as η_e = 1, while the weight for stability-oriented distribution is assigned as η_s = 0.

When both β and $\dot{\beta }$ reside within the extension domain, the measurement mode is set to $M_2=\left\{s\right|0\le K(s)<1\}$. Consequently, the torque distribution weight for energy efficiency optimization is assigned as η_e = K(s), while the weight for stability-oriented distribution is assigned as η_s = 1 − K(s).

When both β and $\dot{\beta }$ reside within the classical domain, the measurement mode is set to $M_3=\left\{s\right|K(s)<-1\}$. Consequently, the torque distribution weight for energy efficiency optimization is assigned as η_e = 0, while the weight for stability-oriented distribution is assigned as η_s = 1.

2.4.4. Objective Function and Constraints for Extenics Coordinated Control

Based on the objective functions and constraints defined in Section 2.4.1 and Section 2.4.2, and incorporating the weighting coefficients obtained in Section 2.4.3, the extenics coordinated control objective function for the hub motor torque is derived as follows:

$$J=\left\{\begin{array}{l}{J}_E\\ {\eta }_e{J}_E+{\eta }_s{J}_s\\ J_s\end{array}\right.\hspace{1em}\begin{array}{c}\left(\beta ,\dot{\beta }\right)\in classical domain\\ \left(\beta ,\dot{\beta }\right)\in extension domain\\ \left(\beta ,\dot{\beta }\right)\in non-domain\end{array}$$

(38)

3. Results and Discussion

Using the proposed in-wheel motor torque distribution control method, a co-simulation experiment was conducted using Matlab/Simulink and Carsim under the NEDC driving cycle and double-lane-change maneuver conditions. Comparative results analysis was performed on the vehicle’s actual motion trajectory and power loss during operation. The main parameters of the vehicle are shown in

Table 3. The main vehicle parameters.

Parameters	Unit	Value	Parameters	Unit	Value
m	kg	1750	Bf	m	1.55
Iz	kg·m2	2954	Br	m	1.55
Lf	m	1.27	Rij	m	0.357
Lr	m	1.4	hg	m	0.54

.

3.1. NEDC Urban Driving Cycle

Assuming the vehicle is traveling on a horizontal road with a road adhesion coefficient of 0.85, the vehicle speed, speed error, in-wheel motor torques, weight coefficients of the objective function, and energy loss are presented in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13.

The desired vehicle speed and actual vehicle speed under three torque distribution methods are shown in Figure 9a. Throughout the driving cycle, all three torque distribution strategies—energy-optimal, stability-oriented, and extenics coordinated control—effectively track the desired speed. As shown in Figure 9b, the energy-optimal distribution exhibits a maximum relative speed error of 21.5%, while the stability-oriented and extenics coordinated control distributions show significantly smaller maximum relative errors of 0.73% and 0.89%, respectively. These results indicate that the energy-optimal distribution performs the worst in longitudinal speed control, followed by the extenics coordinated control distribution, though it only marginally underperforms the stability-oriented approach by 0.16%.

The simulation result curves of in-wheel motor torque for the three distribution control strategies are shown in Figure 10, the weight coefficients for the extenics coordinated control distribution are presented in Figure 11, and the energy losses throughout the driving cycle are illustrated in Figure 12. Under the NEDC urban driving cycle, the total energy loss across all wheels with the energy-optimal distribution is approximately 5.2 × 10⁹ J, while the stability-oriented distribution results in about 5.52 × 10⁹ J total energy loss. The extenics coordinated control distribution demonstrates a total energy loss of approximately 5.4 × 10⁹ J, representing a 3.85% increase compared to the energy-optimal distribution but a 2.17% reduction relative to the stability-oriented distribution.

Therefore, under the NEDC urban driving cycle, when comprehensively considering vehicle speed and energy loss in distributed-drive electric vehicles, the extenics coordinated control-based torque distribution method emerges as the optimal choice compared to both the stability-oriented and energy-optimal torque distribution approaches.

3.2. Double-Lane-Change Maneuver Driving Conditions

Assuming the vehicle travels on a horizontal road at a constant speed of 80 km/h with a road adhesion coefficient of 0.85, the actual vehicle state curves are shown in Figure 13. As shown in Figure 13a,b, the energy-optimal distribution demonstrates significantly larger errors in tracking desired lateral displacement and vehicle speed compared to both the extenics coordinated control and stability-oriented distributions, with the maximum speed error reaching 1.34 km/h. The yaw rate and sideslip angle curves presented in Figure 13c,d reveal that the stability-oriented distribution achieves the smallest errors, while the extenics coordinated control distribution performs slightly worse than the stability-oriented approach. The energy-optimal distribution exhibits a maximum yaw rate error of 2.9 °/s and a maximum sideslip angle error of 1.07°. In comparison, the extenics coordinated control distribution demonstrates a yaw rate maximum relative error of 1.06 °/s (relative error: 5.28%) and a maximum sideslip angle error of 1.0045°. The $\beta -\dot{\beta }$ phase plane trajectory in Figure 13e indicates that the energy-optimal distribution results in the poorest vehicle stability, while Figure 13f shows that the maximum vehicle roll angle reaches 0.22°.

The simulation curves of the three in-wheel motor torque distribution control strategies are shown in Figure 14. The weight coefficients for the extenics coordinated control distribution are presented in Figure 15, indicating that the strategy prioritizes energy-optimal distribution during straight-line driving and shifts to stability-oriented distribution during steering maneuvers. The energy losses throughout the driving cycle are illustrated in Figure 16. Among the three distribution strategies, the stability-oriented distribution demonstrates higher energy losses at each wheel compared to the other two strategies, with the highest total energy loss of 10.92 × 10⁷ J. The energy-optimal distribution achieves the lowest total energy loss of 6.96 × 10⁷ J. The extenics coordinated control strategy shows an intermediate total energy loss of 9.7 × 10⁷ J, which is 28% higher than the energy-optimal distribution but 11.2% lower than the stability-oriented distribution.

Under double-lane-change maneuver driving conditions, analysis of vehicle state parameter response curves, motor torque, and energy loss demonstrates that the extenics coordinated control distribution achieves a significantly superior vehicle stability compared to the energy-optimal distribution, while maintaining a lower energy loss than the stability-oriented distribution. The extenics coordinated control-based torque distribution method effectively combines the advantages of both strategies.

4. Conclusions

This study focuses on distributed-drive electric vehicles and proposes an extensics coordinated torque distribution control method that simultaneously addresses both stability and energy efficiency. A two-degree-of-freedom vehicle dynamics model was established to obtain the reference model for vehicle sideslip angle and yaw rate. A fractional-order PID controller was designed for vehicle speed tracking, while a sliding mode controller was developed for lateral stability control using sideslip angle and yaw rate as control targets. The phase plane was obtained based on different initial states of yaw rate and sideslip angle, with the diamond method and limit cycle method employed to determine the boundaries of the vehicle stability region. An objective function with constraints was formulated with the dual objectives of energy efficiency and vehicle stability, where extension correlation degrees between actual vehicle states and stability region boundaries were calculated using extension control theory to determine weight coefficients for both objectives. A co-simulation platform integrating Matlab/Simulink and Carsim was constructed, and comparative experiments were conducted for three torque distribution methods.

Comparative simulation experiments were conducted under both the NEDC urban driving cycle and double-lane-change conditions, evaluating the following three torque distribution strategies: energy-optimal, stability-oriented, and extenics coordinated control. The co-simulation results demonstrated that under the NEDC urban cycle, the extenics coordinated control strategy achieved a maximum relative speed error of 0.89%, slightly higher than the stability-oriented approach but significantly lower than the energy-optimal distribution. The energy loss with extenics coordinated control measured 5.4 × 10⁹ J, representing a 3.85% increase compared to the energy-optimal distribution but a 2.17% reduction relative to the stability-oriented distribution. Under double-lane-change conditions, the vehicle stability parameters with extenics coordinated control performed slightly worse than the stability-oriented approach but substantially outperformed the energy-optimal strategy. The maximum relative error of yaw rate and the maximum error of sideslip angle for the extenics coordinated control distribution were 5.28% and 1.0045°, respectively. The total energy loss with this method reached 9.7 × 10⁷ J, reflecting a 28% increase compared to the energy-optimal distribution but an 11.2% reduction relative to the stability-oriented approach.

In conclusion, the torque distribution method for distributed-drive electric vehicles based on extenics coordinated control effectively strikes a balance between energy-optimal distribution and stability-oriented distribution. The method not only ensures satisfactory handling stability, but also reduces energy loss in the in-wheel motors, thereby extending the vehicle’s driving range.

Future work will involve automatically converting the algorithm code written in Matlab/Simulink into the required code for the vehicle domain controller via a toolchain, in order to validate the proposed torque distribution method on a real vehicle. Future research will also further consider simulation analysis using a four-wheel vehicle reference model and uneven road conditions.